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## Why speed, distance and time

One of the important topics we come across in Math is Speed, Distance and Time and it is not only limited to Math. You will also learn when you will study Physics. It is one of the most frequently asked topics in competition as well as school exams. So, this is not the topic you might want to skip.

We could find out many real-life examples where calculating speed, distance or time will be useful. For example, you want to find the time to reach your favourite cousin’s place because you are so eager to meet your cousin and travelling in the car is so boring. So, you can check the speed from the speedometer and then calculate the time by dividing it by distance. You will study that formula later. This was one simple example. You can find many such examples by just observing the world around you.

Enough of the talking, let’s get to the real business now.

**Speed**

Speed is defined as how quickly you travel from one place to another. Formally, speed is defined as the distance covered divided by the time taken. i.e.

**Speed = Distance / Time**

The above formula finds the speed. But, the question does not always involve finding speed. We can also be asked for distance or time. Any two of three quantities are given and the remaining one is asked. All of them can be found by the same single formula. However, it gets complex and requires a bit manipulation, for example, to find the time, divide by speed and multiply by time on both sides.

We can easily remember the formulas for all of these quantities using just a triangle method explained below.

## Speed distance time formulas

Let’s see the formula of speed, time and distance one by one.

**What is the formula for speed:**

For finding speed, we can see from the triangle below that distance is above and it is divided by the time (time is below). Therefore,

**Speed = Distance / Time**

**Formula for time**

For finding time, we can observe that distance is above and speed is below which means that distance is divided by speed. i.e.

**Time = distance / Speed**

**What is the formula for Distance**

Lastly, to find the distance, we can see that both speed and time are inline which means that speed and time are multiplied. Therefore,

**Distance = Speed × Time**

We are now going to solve some example problems to understand what we have learnt so far.

### SI units

Note that the speed, distance and time need to be represented along with their units. Without units, they are meaningless. For example, if you say speed is 3. It is meaningless. However, if you say speed is 3 miles/hour, it makes sense. **Speed can be represented by many units. Some of the most common units are listed below:**

**miles/hour (mph)****kilometer/hour (km/hr)****meter/second (m/s)**

Where **mile, kilometer and meter are the units of distance and hour, minute and second are the units of the second.**

Also, note that the units need to be consistent. One of the common mistakes students make is that they do not make the appropriate conversion. For example, if the speed is given in km/hr and distance is given in meters. So, before solving the question, you need to convert it into the proper units. Convert km to meters or vice versa i.e. 1 meter = 1000 km. Inconsistent units will always lead to the wrong answer.

## Time, Speed and distance questions:

**Example 1**

A car travels from Mumbai to Delhi. It needs to cover the distance of 1427 km and the time it takes to reach Delhi is 20 hours. Find the speed?

**Solution**

Car cover distance of 1427 km.

Time taken by the car = 20 hours.

Speed can be found as follows

speed = distance / time

speed = 1427 / 20

speed = 71.35 km/hr

**The car travels at a speed of 71.32 km/hr**

**Example 2**

Ali walks at a speed of 3.1 miles per hour (mph). He has to cover a distance of 0.6 miles to reach his school. How much time it will take Ali to reach the school? Give your answer in minutes.

**Solution**

Speed = 3.1 mph

Distance covered = 0.6 miles

Time can be found as follows

time = distance / speed

time = 0.6 / 3.1

time = 0.1935 h

Since the answer is required in minutes, we need to convert it.

1 hour = 60 minutes. Therefore,

time = 0.1935 × 60

time = 11.61 minutes

**It will take 11.61 minutes to reach school.**

**Example 3**

A motorbike travels at a speed of 40 km/hr for 120 minutes. How many miles it has covered in that amount of time?

**Solution**

Speed = 40 km/hr

Time traveled = 120 minutes

The units are inconsistent here. Before calculating distance, we need to make the appropriate conversion. 1 hour = 60 minutes. Then,

Time traveled = 120 / 60

Time traveled = 2 hours

Now,

distance = speed × time

distance = 40 × 2

distance = 80 km

Answer is required in miles. 1 mile = 1.609 km. Then,

distance = 80 / 1.609

distance = 49.7 miles

**The motorbike has traveled 49.7 miles.**

**Example 4**

A car needs to go from source X to destination Y. It needs to cover the distance of 1400 km and it is moving at a speed of 80 km/hr. Another car starts from the Y after 5 hours and is travelling at a speed of 60 km/hr. At what time will both cars cross each other?

**Solution**

Car 1’s speed = s1 = 80 km/hr

Car 2’s speed = s2 = 60 km/hr

Let’s say car 2 has covered a distance of x when both cars cross one another. The distance covered by car 2 can be found as follows

Distance covered by car 1 + distance covered by car 2 = 1400

Distance covered by car 1 + x = 1400

Distance covered by car 1 = 1400 – x

**For car 1**

Using the formula of distance to get the value of distance in terms of time

distance = speed × time

let time = t

1400 – x = 80t

x = 1400 – 80t

**For car 2**

Using formula to calculate time

time = distance / speed

t = x / 40

substituting the value of x

t = (1400 – 80t) / 40

40t = 1400 – 80t

120t = 1400

t = 1400 / 120

t = 11.67 hours

**Car 1 and Car 2 cross each other after 11.67 hours.**

**Example 5**

A train travels at the speed of 27 mph to cover a distance of 40 miles. If the train travels at the speed of 35 mph. How much additional distance it will cover at the same time?

**Solution**

Initially, train travels at the speed = s1 = 27 mph

distance covered = 40 miles

Let the time taken be t

time = distance / speed

t = 40 / 27

t = 1.48 hours

Speed of the train is changed = s2 = 35 mph

Distance can be found as follows

distance = speed × time

The value of the time was obtained above. Since, both the times are the same. We can use it here

distance = 35 × 1.48

distance = 51.8 miles

We need to find the additional distance so

additional distance = distance covered at new speed – distance covered at an initial speed

additional distance = 51.85 – 40

additional distance = 11.85 miles

**An additional distance of 11.85 miles will be covered.**

**Example 6**

A motorcycle is going at the speed of 3 times the original speed and reaches the destination 2 hours early. What was its original time to reach the same destination?

**Solution**

let the original speed of the motorcycle be s.

New speed = 3s

Let the original time be t hours.

It reaches 2 hours earlier, so new time = t – 2

Let d be the distance covered to reach the destination.

At the initial speed

speed = distance / time

s = d / t

At the changed speed

speed = distance / time

3s = d / (t – 2)

We know that s = d / t from the above formula. Then,

3(d / t) = d / (t – 2)

3 / t = 1 / (t – 2)

3(t – 2) = t

3t – 6 = t

2t = 6

t = 6 / 2

t = 3

**The original time was 3 hours.**

**Example 7**

An aeroplane covers a distance of 3600 km in 5 hours with the wind and covers the same distance in 8 hours against the wind. What is the speed of the plane and the wind?

**Solution**

When an aeroplane moves in the direction of the wind (with the wind), the wind pushes the aeroplane and it helps the plane moves faster than its original speed. Therefore,

Speed of aeroplane with the wind = speed of the plane + speed of the wind

When an aeroplane moves opposite to the direction of the wind (against the wind), the wind hinders the speed of the plane and thus the plane moves slower than its original speed. Therefore,

Speed of aeroplane against the wind = speed of the plane – the speed of the wind

Let the speed of the plane be s.

Let the speed of the wind be w.

speed of the aeroplane with the wind = distance covered with the wind/time taken

speed with the wind = 3600 / 5 = 720 km/hr

Speed of the aeroplane against the wind = distance covered against the wind/time taken

speed against the wind = 3600 / 8 = 450 km/hr

Plugging these values in the formulas for speed with and against the wind. We get,

**720 = s + w → equation (a)**

**450 = s – w → equation (b)**

Adding both these equations. We have,

1170 = 2s

s = 1170 / 2

s = 585 km/hr

**Speed of the plane is 585 km/hr.**

Put the value of s in the equation a or b. Let’s put it into the equation.

720 = s + w

720 = 585 + w

w = 720 – 585

w = 135 km/hr

**Speed of the wind is 135 km/hr.**

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